P, NP, NP-Complete. Ruth Anderson

Transcription

1 P, NP, NP-Complete Ruth Anderson

2 A Few Problems: Euler Circuits Hamiltonian Circuits Intractability: P and NP NP-Complete What now? Today s Agenda 2

3 Try it! Which of these can you draw (trace all edges) without lifting your pencil, drawing each line only once? Can you start and end at the same point? 3

4 Your First Task Your company has to inspect a set of roads between cities by driving over each of them. Driving over the roads costs money (fuel), and there are a lot of roads. Your boss wants you to figure out how to drive over each road exactly once, returning to your starting point. 4

5 Euler Circuits Euler circuit: a path through a graph that visits each edge exactly once and starts and ends at the same vertex Named after Leonhard Euler ( ), who cracked this problem and founded graph theory in 1736 An Euler circuit exists iff the graph is connected and each vertex has even degree (= # of edges on the vertex) 5

6 The Road Inspector: Finding Euler Circuits Given a graph G = (V,E), find an Euler circuit in G Can check if one exists: Check if all vertices have even degree Basic Euler Circuit Algorithm: 1. Do a depth first search from a start vertex until you are back to the start vertex. You never get stuck because of the even degree property. 2. Remove the walk, leaving several components each with the even degree property. Recursively find Euler circuits for these. 3. Splice all these circuits into an Euler circuit Running time? 6

7 The Road Inspector: Finding Euler Circuits Given a graph G = (V,E), find an Euler circuit in G Can check if one exists: (in O( V + E ) ) Check if all vertices have even degree Basic Euler Circuit Algorithm: 1. Do a depth first search from a start vertex until you are back to the start vertex. You never get stuck because of the even degree property. 2. Remove the walk, leaving several components each with the even degree property. Recursively find Euler circuits for these. 3. Splice all these circuits into an Euler circuit Running time? O( V + E ) 7

8 Euler Circuit Example A B C G D E Euler(A) : F 8

9 Euler Circuit Example A B C G D E F Euler(A) : A B G E D G C A 9

10 Euler Circuit Example A B C B C G D E D E F Euler(A) : A B G E D G C A F Euler(B) 10

11 Euler Circuit Example A B C B C G D E D E F Euler(A) : A B G E D G C A F Euler(B): B D F E C B 11

12 Euler Circuit Example A B C B C G D E D E F F Euler(A) : A B G E D G C A Splice A B D F E C B G E D G C A Euler(B): B D F E C B 12

13 Your Second Task Your boss is pleased and assigns you a new task. Your company has to send someone by car to a set of cities. The primary cost is the exorbitant toll going into each city. Your boss wants you to figure out how to drive to each city exactly once, returning in the end to the city of origin. 13

14 Hamiltonian Circuits Euler circuit: A cycle that goes through each edge exactly once Hamiltonian circuit: A cycle that goes through each vertex exactly once Does graph I have: An Euler circuit? A Hamiltonian circuit? Does graph II have: An Euler circuit? A Hamiltonian circuit? B D B D G G C E C E I II 14

15 Finding Hamiltonian Circuits Problem: Find a Hamiltonian circuit in a graph G One solution: Search through all paths to find one that visits each vertex exactly once Can use your favorite graph search algorithm to find paths This is an exhaustive search ( brute force ) algorithm Worst case: need to search all paths How many paths?? 15

16 Analysis of Exhaustive Search Algorithm Worst case: need to search all paths How many paths? Can depict these paths as a search tree: B B D G C E D G C G E D E C G E Etc. Search tree of paths from B 16

17 Analysis of Exhaustive Search Algorithm Let the average branching factor of each node in this tree be b V vertices, each with b branches Total number of paths b b b b Worst case B D G C G E D E C G E Etc. Search tree of paths from B 17

18 Analysis of Exhaustive Search Algorithm Let the average branching factor of each node in this tree be b V vertices, each with b branches Total number of paths b b b b = O(b V ) Worst case Exponential time! B D G C G E D E C G E Etc. Search tree of paths from B 18

19 Running Times 19

20 Time needed to solve problems of various sizes with an algorithm using the indicated number n of bit operations, assuming that each bit operation takes seconds, a reasonable estimate of the time required for a bit operation using the fastest computers available today. Times of more than years are indicated with an asterisk. In the future, these times will decrease as faster computers are developed. From Rosen, Discrete Mathematics and Its Applications,

21 Polynomial vs. Exponential Time All of the algorithms we have discussed in this class have been polynomial time algorithms: Examples: O(log N), O(N), O(N log N), O(N 2 ) Algorithms whose running time is O(N k ) for some k > 0 Exponential time b N is asymptotically worse than any polynomial function N k for any k 21

22 The Complexity Class P P is the set of all problems that can be solved in polynomial time All problems that have some algorithm whose running time is O(N k ) for some k Examples of problems in P: sorting, shortest path, Euler circuit, etc. 22

23 P Sorting Shortest Path Euler Circuit 23

24 P Hamiltonian Circuit Sorting Shortest Path Euler Circuit 24

25 P Sorting Shortest Path Euler Circuit Hamiltonian Circuit Satisfiability (SAT) Vertex Cover Travelling Salesman 25

26 Satisfiability Input: a logic formula of size m containing n variables Output: An assignment of Boolean values to the variables in the formula such that the formula is true O(m*2 n ) algorithm: Try every variable assignment 26

27 Vertex Cover: Input: A graph (V,E) and a number m Output: A subset S of V such that for every edge (u,v) in E, at least one of u or v is in S and S =m (if such an S exists) O(2 m ) algorithm: Try every subset of vertices of size m 27

28 Traveling Salesman Input: A complete weighted graph (V,E) and a number m Output: A circuit that visits each vertex exactly once and has total cost < m if one exists O( V!) algorithm: Try every path, stop if find cheap enough one 28

29 A Glimmer of Hope If given a candidate solution to a problem, we can check if that solution is correct in polynomial-time, then maybe a polynomial-time solution exists? Can we do this with Hamiltonian Circuit? Given a candidate path, is it a Hamiltonian Circuit? 29

30 A Glimmer of Hope If given a candidate solution to a problem, we can check if that solution is correct in polynomial-time, then maybe a polynomial-time solution exists? Can we do this with Hamiltonian Circuit? Given a candidate path, is it a Hamiltonian Circuit? just check if all vertices are visited exactly once in the candidate path 30

31 The Complexity Class NP Definition: NP is the set of all problems for which a given candidate solution can be tested in polynomial time Examples of problems in NP: Hamiltonian circuit: Given a candidate path, can test in linear time if it is a Hamiltonian circuit Satisfiability: Given a circuit made out of AND, OR, NOT gates, and an assignment of values, is the output 1? All problems that are in P (why?) 31

32 NP P Sorting Shortest Path Euler Circuit Hamiltonian Circuit Satisfiability (SAT) Vertex Cover Travelling Salesman 32

33 Why do we call it NP? NP stands for Nondeterministic Polynomial time Why nondeterministic? Corresponds to algorithms that can guess a solution (if it exists), the solution is then verified to be correct in polynomial time Can also think of as allowing a special operation that allows the algorithm to magically guess the right choice at each branch point. Nondeterministic algorithms don t exist purely theoretical idea invented to understand how hard a problem could be 33

34 Your Chance to Win a Turing Award! It is generally believed that P NP, i.e. there are problems in NP that are not in P But no one has been able to show even one such problem! This is the fundamental open problem in theoretical computer science Nearly everyone has given up trying to prove it. Instead, theoreticians prove theorems about what follows once we assume P NP! 34

35 NP-completeness Set of problems in NP that (we are pretty sure) cannot be solved in polynomial time. These are thought of as the hardest problems in the class NP. Interesting fact: If any one NP-complete problem could be solved in polynomial time, then all NP-complete problems could be solved in polynomial time. Even more: If any NP-complete problem is in P, then all of NP is in P 35

36 NP P Sorting Shortest Path Euler Circuit NP-Complete Hamiltonian Circuit Satisfiability (SAT) Vertex Cover Travelling Salesman 36

37 Saving Your Job Try as you might, every solution you come up with for the Hamiltonian Circuit problem runs in exponential time.. You have to report back to your boss. Your options: Keep working Come up with an alternative plan 37

38 In general, what to do with a Hard Problem Your problem seems really hard. If you can transform an NP-complete problem into the one you re trying to solve, then you can stop working on your problem! 38

39 What do we do about it? Approximation Algorithm: Can we get an efficient algorithm that guarantees something close to optimal? Heuristics: Can we get something that seems to work well most of the time? Restrictions: Maybe you have stated your problem too generally. Many hard problems are easy for restricted inputs. 39

40 Great Quick Reference Computers and Intractability: A Guide to the Theory of NP-Completeness, by Michael S. Garey and David S. Johnson 40